Fort Worth already has two sports teams if you count Arlington as a Fort Worth suburb. The problem is, they're not named Fort Worth since Arlington is also a Dallas suburb.
If Fort Worth had a sports team named after it, it would significantly help in getting people to recognize Fort Worth as a metro anchor.
Unfortunately, it's unlikely to ever happen. You'd have to rename a team that's already in the Fort Worth area, and/or attract a team that would compete with another team in the same TV market.
You make a very good point in your second sentence. I think a lot of us miss the point about a city's identify and the role the national media play, particularly as to sporting events. Take Dallas, for instance. I truly believe that, prior to 1963, Dallas was largely viewed across the country as just some city in North Central Texas. Before that time, too, I doubt if many Americans on the east and west coasts cared much of anything about Texas, except their perception of a land of cowboys, Indians, and rich oil millionaires.
Then came two events that put Dallas in the spotlight. The first, and tragic event, was the Kennedy assassination. But what lingered longer in the national public eye was the emergence and growing popularity of the Dallas Cowboys football team. And Dallas's smart marketing of the Cowboys as "America's team" augmented the public perception of Dallas as the home of the Super Bowl-winning Cowboys.
Incidentally, most don't know this, but the regional Associated Press is located in Dallas. So any big news event in our region that is covered by AP gets the location name "Dallas." It also helps to have all the major TV networks' affiliates identifying themselves as being in Dallas. Because Dallas has the major sports franchises that are covered nationally it affects just about everything else regarding that city's identity. Have you ever heard of any news writer or announcer say "Dallas, Texas?" No, they just say "Dallas." Like they would say New York or Chicago. But they say "Fort Worth, Texas." What about San Francisco, San Diego, Oakland, Boston, New Orleans, on and on? You never hear the name of the state, like San Diego, California, or Boston, Massachusetts, or New Orleans, Louisiana. Yes, New Orleans is itself historically unique. But Oakland or Green Bay have no historical significance. The only factor that makes Oakland and Green Bay worth noting nationwide is professional football. And Oakland fans were certainly angry when their pro football franchise left them.
In my opinion, national public perception, augmented by the national news media, creates a city's identify. But, back to the sports rivalry idea, I believe our city would profit immensely, money-wise and in terms of national identify, if we have a major sports team to rival Dallas's.